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Compound Interest Formula Derivation, CI Example Quarterly/Half Yearly in Excel

Compound Interest Formula

The interest is added back to the principal sum in order that interest can be earned on that additional interest during the next cyclical period. To understand the Compound Interest Formula let’s take a look at this article where the CI Formula along with derivation furnished.  The basic the conception as well as Compound Interest Formula for Quarterly/Half Yearly and also its illustrations provided on this page.

The major benefit of Compound Interest Formula is that Compound Interest Formula for Excel is used for debit as well as credit aspects in the terms of financial world. Compound Interest Formula is also used in Excel or Java. To know more about the Compound Interest Formula Derivation and its uses, individuals need to scroll down this page which is well prepared by the Dedicated and inspired team of recruitmentresult.com

Compound Interest Formula

Formula Of Compound Interest

The formula for annual compound interest, including principal sum, is:

Compound Interest Formula: A = P (1 + r/n) (nt)

Where:

  • A = amount of money accumulated after n years, including interest.
  • P = principal amount (the initial amount you borrow or deposit)
  • r = annual rate of interest (as a decimal)
  • t = number of years the amount is deposited or borrowed for.
  • n = number of times the interest is compounded per year

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Fundamental Compound Interest Formulas

Compound Interest Formula For Monthly:

Amount = P (1 + R / 100) n

Compound Interest Formula For Half-yearly:

Amount = P (1 + (R/2)/ 100) 2n

Compound Interest Formula For Quarterly:

Amount = P (1 + (R/4)/ 100)4n

When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd and 3rd year respectively:

Amount = P (1 + R 1 / 100) (1 + R 2 / 100) (1 + R 3 / 100)

Present worth of Rs. x due n years hence is given by:

 Present Worth = X / (1 + R / 100)

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Compound Interest Formula Derivation

How to calculate the Compound Interest?

Write P for your starting principal, and r for the rate of return expressed as a decimal to find out the formula for future value (So if the interest rate is 5%, r equals .05). To know your balance go through the following table;

YearBalance
NowP
1P + rP
2(P + rP) + r(P + rP)

This starts to get messy in a hurry. But you can simplify it by noticing that you can keep pulling out factors of (1 + r) from each line. If you do that, the balances collapse to a simple pattern:

YearBalance
NowP
1P(1 + r)
2P(1 + r)2

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How to calculate compound interest In Math?

Calculating the Total Value of the Compound Interest Formula For Deposit

P (1+ i/n)nt

Step 1: 10,000 (1+0.05/4)4×10

Step 2: 10,000(1+0.0125)40

Step 3: 10,000 (1.0125)40

Step 4: 10,000 (1.64361946349)

Step 5: 16436.1946349

We can round of this total to Rs. 16,436.19. So the compound interest earned after 10 years is Rs. 6,436.19.

Calculating the Interest Earned

We can also arrive at this figure using the formula for compound interest earned. We can substitute the numbers for letters as seen below:

P [(1+ i/n)nt -1]

Step 1: 10,000 [(1+0.05/4)4×10 -1]

Step 2: 10,000 [(1+0.0125)40-1]

Step 3: 10,000 [(1.0125)40-1]

Step 4: 10,000 [(1.64361946349) -1]

Step 5: 10,000 (0.664361946349

Step 5: 6436.1946349

We can now add this interest earned to the principal amount to find out the value of the deposit. The maturity value will be Rs. 16,436.19.

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Compound Interest Formula in Excel

How to calculate Compound Interest in Excel

Generic formula

=FV (rate,nper,pmt,pv)

Explanation

To calculate compound interest in Excel, you can use the FV function.

This example assumes that $1000 is invested for 10 years at an annual interest rate of 5%, compounded monthly.

In the example shown, the formula in C10 is:

=FV (C6/C8,C7*C8,0,-C5)

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Compound Interest Formula Derivation

Annual Compound Interest Formula

To calculate the compound interest for a number of years together, we need to multiply P(1+i) to the power of the number of years of the deposit. So we end up with this formula:

P (1+ i/n)n

Compound Interest Formula Half-Yearly, Quarterly, Monthly

If you are earning interest multiple times in a year, you need to factor in this number into the equation. So the formula generated is:

P (1+ i/n)nt

This formula can also be used for instances where the interest is compounded once every two years. In this case, n = 0.5, as each year is calculated as half.

Compound Interest Formula In Java

How to calculate compound interest In Java?

In simple, compound interest is a interest on interest, It’s the result of reinvesting.

The formula to find the compound interest

A=P (1 + r/n)nt

Where,

A =” Ending Amount “

P =” Principal “

R =” Interest Rate “

N =” Number of compounding a year “

T =” Total Number Of Years “

Java Program to calculate Compound Interest

We take principal (p), time (t), rate of interest (r) and the number of times the interest is compound (n) as inputs. Amount is calculated using the formuale p * ( 1 + r/n )nt. In Java, the Math.pow() function is used to calculate powers. For example, to calculate 32, we use Math.pow(3,2). After finding amount, we find interest by subtracting principal.

Sample Execution:

Input:
p = 3400
t = 3
r = 4
n = 2

Output:

Compound Interest is 2475200.0

Amount is 2478600.0

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Compound Interest Formula Derivations

The basic formula for Compound Interest is: FV = PV (1+r)n

FV = Future Value,

PV = Present Value,

r = Interest Rate (as a decimal value), and

n = Number of Periods

FV = PV (1+r)n

Find the Future Value when we know a Present Value, the Interest Rate and number of Periods.

PV = FV / (1+r)nFind the Present Value when we know a Future Value, the Interest Rate and number of Periods.
r = ( FV / PV )1/n – 1Find the Interest Rate when we know the Present Value, Future Value and number of Periods.
n = ln (FV / PV)ln(1 + r)

Find the number of Periods when we know the Present Value, Future Value and Interest Rate

How to Calculate Compound Interest on Recurring Deposit?

When it comes to Recurring Deposits, interest amount is compounded every quarter. This is then added up and the final amount that customers receive can be determined. The formula used to calculate compound interest is as follows-

A = P (1 + r/n) nt

For example,

Sita has made an initial investment of Rs.1 lakh for a period of 5 years. The interest rate applicable is 8%. By using the above formula, the final amount that he will get is Rs.1.5 lakh.

START Your Practice With Online: Mathematics Quiz

How to Calculate Recurring Deposit Maturity Amount & Interest?

Calculating Recurring Deposit Maturity Amount:

Now, we will calculate amount (A) from above formula for each installment we pay, starting from first month to 12th month and then add all of them. So the final maturity amount will be

A = A1+A2+A3+…..+A12

A1, A2, A3,,,A12 are the maturity amount for respective installment

Compound Interest Equations

Compound Interest Equation and Calculation

PeriodDeposit Balance
InvestmentP
Year 1P + iP
Year 2(P+ iP) + i(P+iP)

To collapse this formula, we can pull out factors of (1+i). Simply substitute iP with (1+i) to get the following:

PeriodDeposit Balance
InvestmentP
Year 1P(1+i)
Year 2P(1+i)2
Year 3P(1+i)3

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Compound Interest Calculator

Simple Interest vs. Compound Interest

PeriodDeposit 1 – Compound InterestDeposit 2 – Simple InterestDifference
Year 1Rs. 500Rs. 500Rs. 0
Year 2Rs. 1,025.00Rs. 1,000Rs. 25
Year 3Rs. 1,576.25Rs. 1,500Rs. 76.25
Year 4Rs. 2,115.06Rs. 2,000Rs. 115.06
Year 5Rs. 2,762.82Rs. 2,500Rs. 762.82
Year 6Rs. 3,400.96Rs. 3,000Rs. 400.96
Year 7Rs. 4,071.00Rs. 3,500Rs. 571.00
Year 8Rs. 4,774.55Rs. 4,000Rs. 774.55
Year 9Rs. 5,513.28Rs. 4,500Rs. 1,013.28
Year 10Rs. 6,288.95Rs. 5,000Rs. 1,288.95

Compound Interest with Monthly Contributions

PeriodInvestment BreakdownInvestment + Interest AccumulatedInterest EarnedTotal Value of Deposit
Year 110,000 + 12,00022,0001,10023,100
Year 210000 + (12000 x 2) + 1,10035,1001,75536,855
Year 310000 + (12000 x 3) + (1,100 +1,755)48,8552,442.7551,297.75
Year 410000 + (12000 x 4) + (1,100 +1,755 + 2,442.75 )63.297.753164.8766,462.64
Year 510000 + (12000 x 5) + (1,100 +1,755 + 2,442.75 + 3164.87 )78,461.753,923.1382,385.77

Compound Interest Formula With Example

Here we have provided Compound Interest Formula Shortcut with essential Compound Interest Formula illustrations. Have a look here;

Example -1

If an amount of $5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, the value of the investment after 10 years can be calculated as follows…

Solution

P = 5000. r = 5/100 = 0.05 (decimal). n = 12. t = 10.

If we plug those figures into the formula, we get:

A = 5000 (1 + 0.05 / 12) ^ (12(10)) = 8235.05.

So, the investment balance after 10 years is $8,235.05.

Example -2

An investment earns 3% compounded monthly. Find the value of an initial investment of $5,000 after 6 years.

Solution

Determine what values are given and what values you need to find.

Earns 3% compounded monthly: the rate is r=0.03r=0.03 and the number of times compounded each year is m=12m=12

Initial investment of $5,000: the initial amount is the principal, P=5000P=5000

6 years: t=6t=6

You are trying to find AA, the future value (the value after 6 years). Now apply the formula with the known values:

A = P (1 + r/m) (mt)

= 500(1 + 0.03/12) 12 * 6

= 5984.74

Example -3

Find the amount of $ 8000 for 3 years, compounded annually at 5% per annum. Also, find the compound interest.

Solution

Here, P = $ 8000, R = 5 % per annum and n = 3 years.

Using the formula A = $ P(1 + R/ 100)ⁿ

amount after 3 years = $ {8000 × (1 + 5/100)³}

= $ (8000 × 21/20 × 21/20 × 21/20)

= $ 9261.

Thus, amount after 3 years = $ 9261.

And, compound interest = $ (9261 – 8000)

Therefore, compound interest = $ 1261.

Example -4

Find the compound interest on $ 6400 for 2 years, compounded annually at 7¹/₂ % per annum.

Solution

Here, P = $ 6400, R % p. a. and n = 2 years.

Using the formula A = P (1 + R/100)ⁿ

Amount after 2 years = [6400 × {1 + 15/(2 × 100)}²]

= $ (6400 × 43/40 × 43/40)

=$ 7396.

Thus, amount = $ 7396

and compound interest = $ (7396 – 6400)

Therefore, compound interest = $ 996.

Example -5

Find the amount of $ 12000 after 2 years, compounded annually; the rate of interest being 5 % p.a. during the first year and 6 % p.a. during the second year. Also, find the compound interest.

Solution

Here, P = $12000, p = 5 % p.a. and q = 6 % p.a.

Using the formula A = {P × (1 + P/100) × (1 + q/100)}

amount after 2 years = $ {12000 × (1 + 5/100) × (1 + 6/100)}

= $ (12000 × 21/20 × 53/50)

=$ 13356

Thus, amount after 2 years = $ 13356

And, compound interest = $ (13356 – 12000)

Therefore, compound interest = $ 1356.

Benefits of Compound Interest Formula

Compound interest is used for both debit and credit aspects of the financial world. Listed below are some of the investments and credit options that use compound interest. In the field of Investments Compound Interest Formula is used for Savings Accounts, Fixed Deposits, Recurring Deposits, Other Certificates of Deposits, Reinvested Dividend Stocks, and Retirement Funds and also for Debt as Loans, Credit Cards, and Mortgages.

Note

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